Partial Integrated Guidance and Control of

Surface-to-Air Interceptors for High Speed Targets

Radhakant Padhi, Charu Chawla, Priya G. Das and Abhiram Venkatesh

Abstract— An important limitation of the existing IGC al-gorithms, is that they do not explicitly exploit the inherenttime scale separation that exist in aerospace vehicles betweenrotational and translational motions and hence can be inef-fective. To address this issue, a two-loop partial integratedguidance and control (PIGC) scheme has been proposed in thispaper. In this design, the outer loop uses a recently developed,computationally efficient, optimal control formulation named asmodel predictive static programming. It gives the commandedpitch and yaw rates whereas necessary roll-rate command isgenerated from a roll-stabilization loop. The inner loop tracksthe outer loop commands using the Dynamic inversion philos-ophy. Uncommonly, Six-Degree of freedom (Six-DOF) model isused directly in both the loops. This intelligent manipulationpreserves the inherent time scale separation property betweenthe translational and rotational dynamics, and hence overcomesthe deficiency of current IGC designs, while preserving itsbenefits. Comparative studies of PIGC with one loop IGC andconventional three loop design were carried out for engagingincoming high speed target. Simulation studies demonstrate theusefulness of this method.

Keywords- partial integrated guidance and control, highspeed targets, model predictive static programming, dynamicinversion

I. INTRODUCTION

Stringent performance demands are expected for new-

generation interceptors to engage high speed targets, espe-

cially in defense applications. This is a challenging problem

because of several reasons, which include high relative

velocity (and hence very less time availability), high altitude

engagement and hence limited turning capability (due to

reduced dynamic pressure), hit-to-kill capability (i.e. zero

miss distance) demand etc. The trajectory of an interceptor

is typically divided into three segments consisting of lift-

off, mid-course and terminal phases respectively [1]. Out of

these three phases, the terminal homing is a very crucial

phase. In this phase the interceptor locks on to the target

directly (through a seeker) and the guidance and control

algorithms are designed in such a way that the residual errors

of the prior phases get corrected to achieve the minimum

miss distance. However, conventional separated guidance and

control design may not achieve this objective.

Radhakant Padhi, Asst. Professor (Contact author), is with the Departmentof Aerospace Engineering, Indian Institute of Science, Bangalore, India.Email: padhi@aero.iisc.ernet.in

Charu Chawla and Priya G. Das, Graduate students, are with the De-partment of Aerospace Engineering, Indian Institute of Science, Bangalore,India.

Abhiram Venkatesh, Formerly Project Asst., was with the Department ofAerospace Engineering, Indian Institute of Science, Bangalore, India.

The conventional guidance and control philosophy works

in three loops. In the outermost loop, a point mass engage-

ment model is considered to generate lateral acceleration

(Latax) commands for achieving minimum miss distance.

These latax commands are then converted into equivalent

body rates in an intermediate loop, which are tracked by

the inner most loop [2]. However, whenever a loop structure

is followed, it introduces time lags between the inner and

outer loops. In case of conventional engagements the target

usually moves slowly and hence the time availability is

relatively large. In such a situation this three-loop structure

usually turns out to be sufficient. However, in case of

incoming high speed targets (like ballistic missiles) the time

availability is quite less (especially in the terminal phase).

Both the guidance and autopilot loops are decoupled and

hence it achieves lethality in a limited sense. Moreover,

three loop design may demand higher maneuverability, which

subsequently may lead to control saturation.

To overcome the deficiencies of the conventional three

loop design, integrated guidance and control design (IGC)

ideas have been proposed in the recent literature. In this

design philosophy, attempt is made to achieve the objectives

using the full nonlinear Six-DOF interceptor dynamics in

a single unified (single loop) framework. Techniques like

feedback linearization [2], State Dependent Riccati Equation

(SDRE) technique [3], etc. have been applied in the IGC

framework.

Existing IGC algorithms do not explicitly exploit the

inherent time scale separation that exist in aerospace vehicles

between rotational and translational motions. Hence, such an

algorithm can become ineffective unless the engagement ge-

ometry is close to the collision triangle. From our numerical

experiments on published ideas of IGC, we observed that

the control surface deflections directly respond to the transla-

tional error correction demands, which leads to instability of

the rotational dynamics. A close examination reveals that the

control surface deflections (which are located at the tail of the

interceptor) can create only minor forces, whereas they create

large moments due to a long moment arms from the center

of gravity. Hence, they are very ineffective in translational

error correction directly, whereas they can be very effective

in turning the vehicle. Hence for any guidance and control

algorithm (including IGC) to be successful, one must exploit

the time scale separation that exist in aerospace vehicles

between faster rotational dynamics and slower translational

dynamics.

To address the deficiency of existing IGC designs, a two-

loop partial integrated guidance and control (PIGC) structure

2009 American Control ConferenceHyatt Regency Riverfront, St. Louis, MO, USAJune 10-12, 2009

FrA07.5

978-1-4244-4524-0/09/$25.00 ©2009 AACC 4184

is proposed in this paper. In this design, the commanded

pitch and yaw rates are directly generated from an outer

loop optimal control formulation, which is solved in a com-

putationally efficient manner using the recently-developed

model predictive static programming (MPSP) technique [4].

The necessary roll-rate command is generated from a roll-

stabilization loop. The inner loop generates the necessary

control using the Dynamic inversion (DI) philosophy. Un-

commonly, in both the loops the Six-DOF interceptor model

is used directly. Six-DOF simulation studies have been

carried out in three dimentional environment to demonstrate

the usefulness of this method.

II. SYSTEM MODEL

The whole system is defined by Six-DOF interceptor

model and 3-DOF target model. The equations are derived

in fin and inertial frame.

A. Interceptor Model

A general nonlinear system of the interceptor can be

defined as

X = f (X ,Uδ ) (1)

where

X , [u v w p q r xm ym zm xm ym zm q1 q2 q3 q4 ζ ]T (2)

Uδ , [δr δp δy]T (3)

The elaborated plant model can be written in the following

form:

u

v

w

=

vr−wq− QSCDm

−Q11g

wp−ur + QSCNBNm

− QSCNδδy

m− (Q21+Q31)g√

2

uq− vp+ QSCNANm

− QSCNδδp

m− (Q31−Q21)g√

2

(4)

p

q

r

=

1IXX

(

−QSdCRM−QSdClξ

δr +QSd2CLP

p

2Vm

)

1IYY

(

QSdCMXCGA −QSCNδ(XCPδ

−XCG)δp

−(IXX − IZZ)pr +QSd2CMQ

q

2Vm

)

1IZZ

(

QSdCMXCGB −QSCNδ(XCPδ

−XCG)δy

−(IYY − IXX )pq+QSd2CNR

r

2Vm

)

(5)

q1

q2

q3

q4

=

12(q4 p−q3q+q2r)

12(q3 p−q4q+q1r)

12(−q2 p+q1q+q4r)

12(−q1 p−q2q−q3r)

(6)

ζ = p (7)

where, u,v,w are the interceptor velocity components and

p,q,r are the actual body rates both defined in fin frame.

ζ is the roll angle. q1,q2,q3,q4 are the quaternions which

defines the attitude of the interceptor [1]. xm,ym,zm defines

the position coordinates of the interceptor in the inertial

frame. The position rates in inertial frame are given by

xm

ym

zm

= T−1IB

ub

vb

wb

(8)

In (8) ub,vb,wb are the interceptor velocity components in

body frame and TIB is the time varying transformation matrix

between inertial and body coordinate system, given by

TIB =

Q11 Q12 Q13

Q21 Q22 Q23

Q31 Q32 Q33

(9)

where the elements of the matrix are obtained through

quaternions (q1,q2,q3,q4) [1]. The inertial acceleration of

the interceptor is given by

xm

ym

zm

= T−1IF

u− (vr−wq)v− (wp−ur)w− (uq− vp)

(10)

where TIF = TBF TIB is the transformation matrix from inertial

to fin frame. In this transformation, TBF is the constant

transformation matrix between body frame to fin frame. Fin

frame is achieved by a π4

rotation of body frame along

positive Xb-axis, pointing in the direction of nose of the

interceptor. Some of the parameters used in the model are

Q, the dynamic pressure, Vm, the total interceptor velocity,

S, the reference area, m, the mass of the interceptor, d, the

diameter, g, the acceleration due to gravity and Ixx, Iyy, Izz

the moment of inertia about body axes respectively. The

aerodynamic force and moment coefficients are evaluated in

fin frame. CD is the drag coefficient, CNAN is the pitch and

CNBN is the yaw force coefficient respectively. CNδ is the tail

normal force coefficient per unit δ per pair, CRMis the rolling

moment coefficient, Clξis the roll moment control coefficient

per unit roll deflection, CLP,CMQ

and CNRare damping

derivatives, CMXCGA and CMXCGB are pitching and yawing

moment coefficients respectively. XCPδis the tail moment

arm (with respect to nose), XCG is the axial position of center

of gravity from nose (in meters). The necessary control Uδ

is generated through fin (control surfaces) deflections.

B. Target Model

The 3-DOF model of target in inertial frame based on flat

earth assumptions is used. Target is not maneuvering and

hence follows a straight line flight path.

Vt

ψt

γt

xIt

yIt

zIt

=

−Dmt

−gsin(γt)

0−gcos(γt )

Vt

Vtsin(γt)Vtsin(ψt)cos(γt)Vtcos(γt)cos(ψt)

(11)

where D is the drag (aerodynamic force), mt is the mass

of the target, Vt is the total velocity of the target, γt is the

flight path angle of the target, ψt is the heading angle of the

target and xIt yI

t zIt are the position coordinates of the target

in inertial frame.

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C. Design Philosophy of PIGC

In PIGC, the guidance loop uses the translational dynamics

to generate the commanded pitch and yaw rates which

serves the command for the DI loop. The necessary roll-

rate command is generated from a roll-stabilization loop. In

DI loop, the commanded body rates are tracked to generate

the necessary fin deflections which will drive in meeting

the objective of minimum miss distance as shown in Fig.1.

The objective of almost zero miss distance is achieved by

Body RateCommand Generation(MPSP-DI)

Fin Deflection Generation (DI)

Missile

Computation

Target States

u, v, w, p, q1, q

2, q

3, q

4, xrm, yrm, zrm

p, q, r

p*

q* r*

δ δδr p yp*

ζ

Fig. 1. Block Diagram of PIGC scheme

minimizing the relative error in position and velocity in y

and z direction in fin frame with no control authority in X

direction. To visualize the combined effect of guidance and

I

x

y

Z

Interceptor

I

Target

LOS

o o'

ffI

fI

Inertial Frame

x y zI II

Rotating fin frame coordinates

I zI

f ff

m

m

m

y

xI

zI

x y I

x

yz

Fig. 2. Engagement Geometry in Rotating fin frame

control, transition between three reference frames namely,

body frame, inertial frame and fin frame is considered in

the design. The origin of the body coordinate frame is

assumed to be at the interceptor center of gravity. The inertial

coordinates are defined by earth fixed inertial frame. Since

the interceptor seeker defines the target position relative to

the interceptor body coordinates, it is desirable to define

the interceptor and target position in a coordinate system

parallel to the instantaneous interceptor fin axes with the

origin coinciding with the origin of the inertial frame as

shown in Fig.2. The rotating coordinate system is defined

as [x fI , y

fI , z

fI ] and the inertial frame is defined as [xI , yI , zI ].

The interceptor fin frame fixed on the interceptor body is

defined as [xm, ym, zm]. The relative positions of the target

with respect to interceptor in the rotating frame is given by

Xrel , [xrm yrm zrm]T (12)

The relative velocity between interceptor and target is given

by

xrm

yrm

zrm

= TIF

xIt

yIt

zIt

−

u

v

w

−

qzrm − ryrm

rxrm − pzrm

pyrm −qxrm

(13)

Equation (13) is obtained by differentiating (12) such that

derivative of the transformation matrix TIF is given by [5]:

TIF = −ω TIF , ω =

0 −r q

r 0 −p

−q p 0

(14)

where ω is the skew-symmetric matrix.

III. PIGC: MATHEMATICAL DETAILS

This section describes the implementation scheme of the

PIGC.

A. Guidance (Outer Loop): MPSP

1) Generic Theory: In the MPSP technique, one needs

to start from a “guess history” of the control solution. With

the control guess history, obviously the objective will not

meet, and hence, there is a need to improve the solution. In

this section, we present a way to compute an error history

of the control variable, which needs to be subtracted from

the previous history to get an improved control history. A

general nonlinear system is defined as

X = f (X ,U)

The discretized form of the state and output dynamics are

Xk+1 = Fk(Xk,Uk), Yk = h(Xk) (15)

where X ∈ ℜn, U ∈ ℜm, Y ∈ ℜp and k = 1,2, . . . ,N are the

time steps.

Xk+1 = Xk +∆t( f (X ,U)) (16)

However, from (15), we can write the error in state at time

step (k +1) as

dXk+1 =

[

∂Fk

∂Xk

]

dXk +

[

∂Fk

∂Uk

]

dUk (17)

where dXk and dUk are the error of state and control at time

step k respectively. The respective values for∂Fk

∂Xkand

∂Fk

∂Uk

are obtained from the discretized state model of (16). We

aim for the output vector YN → Y ∗N . Expanding YN about

Y ∗N using Taylor series expansion and then using small error

approximation we can write the error in the output, ∆YN ,

YN −Y ∗N as△YN

∼= dYN =[

∂YN

∂XN

]

dXN . The error in output

obtained as:

dYN =

[

∂YN

∂XN

]([

∂FN−1

∂XN−1

]

dXN−1 +

[

∂FN−1

∂UN−1

]

dUN−1

)

(18)

Similarly the error in state at time step (N −1), dXN−1 can

be expanded in terms of dXN−2 and dUN−2 and so on.

Continuing the process until k = 1, we obtain

dYN = B1dU1 +B2dU2 + . . .+BN−1dUN−1 =N−1

∑k=1

BkdUk

(19)

4186

assuming the error in the initial conditions is zero and it

means dX1 = 0. The key point is that the sensitivity matrices

Bk is computed recursively which will save the computational

time enormously [4]. In (19), we have (N −1)m unknowns

and p equations. Usually p << (N−1)m and hence, it is an

under-constrained system of equations. Hence, an additional

objective is met by minimizing the following objective (cost

function) subjected to the constraint in (19)

J =1

2

N−1

∑k=1

(U0k −dUk)

T Rk(U0k −dUk)

Here, U0k , k = 1, . . . ,(N −1) is the previous control history

solution and dUk is the corresponding error in the control

history. Rk is a positive definite matrices which is chosen

judiciously by the control designer. After carrying out the

necessary algebra [7] we get

dUk = −R−1k BT

k A−1λ (dYN −bλ )+U0

k (20)

Hence, the updated control at time step k = 1,2, . . . ,(N −1)is given by

Uk = U0k −dUk (21)

One can refer to the details of MPSP technique in [4]

2) Mathematical Formulation: MPSP starts with the

initial guess history which is generated from Propor-

tional Navigation (PN) law off line. Curve fitting of the

initial guess history obtained through PN law is done

for the present study. It also provides the value for fi-

nal time/time-to-go (t f ). MPSP algorithm works in two

modes, namely the correction and prediction mode. In

the correction mode, 9 states are considered given as

X ,[

v w q1 q2 q3 q4 xrm yrm zrm

]TEqua-

tions (4), (5), (6) and (13) are discretized using Euler

Integration to get the discretized model [6]. Hence, the size

of the matrix∂Fk

∂ Xkobtained in the present work is 9 by 9.

The control for the outer loop formulation is the commanded

pitch and yaw rate U ,[

q∗ r∗]T

. Hence, the size of the

matrix∂Fk

∂Ukobtained is 9 by 2. The output of the MPSP

is YN ,[

yrm zrm yrm zrm

]T. The desired output is

Y ∗N ,

[

0 0 0 0]T

. In the prediction mode 17 states

are considered as stated in (2) to predict the actual plant

model. The convergence is obtained when the relative error

in position and velocity of target and interceptor in yfI and

zfI direction in fin frame as shown in Fig.2 decreases to a

minimum specified value at t f . The fin deflections are treated

as time varying parameters in the correction mode of the

MPSP formulation.

B. Control(Inner Loop) : Dynamic Inversion

The control loop is based on DI which ensures a good

command tracking [8]. The fin deflection serves as the con-

trol for the inner loop and stabilizes the rotational dynamics.

The error in commanded body rates and the actual body

rates will generate fin deflections by enforcing the error

dynamics. The commanded body rates are obtained from the

guidance (MPSP) loop. The output vector for the inner loop

is defined as Y =[

p q r]T

and the control vector is

Uδ =[

δr δp δy

]T. The governing state model for inner

loop contains only the body rates. The DI formulation results

in a square system (no. of outputs same as number of inputs)

[8]. The control explicitly appears in the first order derivative

of the output variable in the inner loop. Thus,

Y = fy(X)+gy(X)U (22)

The goal is Y → Y ∗, where Y ∗ = [p∗ q∗ r∗]T . The error

of tracking is defined as E , Y (t)−Y ∗(t). Enforcing the

first order error dynamics, such that E → 0 as t → ∞. Here,

ki = diag( 1τ ) and τ = Ts/4 is the time constant of the first

order system. It is selected based on the small settling time

(Ts = 0.4sec) due to the fast nature of the rotational dynamics.

Uδ = [gy(X)]−1[−( fy(X)− Y ∗)−K(Y −Y ∗)] (23)

The updated value of fin deflections Uδ is used for prop-

agation of system dynamics. The inner loop assures the

convergence of the body rates to their desired/commanded

values obtained from the guidance loop such that q →q∗ and r → r∗. In this way, the rotational dynamics will

only be responsible for generating the effective control (fin

deflections) and no over coupling with translational dynamics

is accounted which was happening earlier in IGC design.

IV. SIMULATION RESULTS

For our simulation studies, a surface to air interceptor

is considered which is aiming for a high speed threat in

terminal homing. Both the target and interceptor model

equations are simulated by using Runge-Kutta method to

get the required knowledge of the updated plant dynamics

trajectory [6]. The guidance cycle is 12.5msec and the state

update cycle is 2.5msec. The time of engagement (t f ) is

3.9973sec. Assuming the lethal range of 5m, we got the final

miss distance less than 0.5m. Results appears to be promising

in context with the zero effort miss (ZEM) which is around

500m. It is the distance by which the interceptor will miss

the target if it continues along its present course with no

control applied by the interceptor.

A. Comparison between Three-Loop (Conventional) Design

and PIGC

In conventional systems, guidance and control subsystems

are designed separately and then integrated together into the

missile. The performance of such systems are not optimized

truely. The logic of PIGC is validated by carrying out the

comparative study with the conventional design. We can

observe from the Fig. 3, that the commanded body rates has

a straight line profile compared to that of PN law used in

the conventional design. This comes with the advantage of

natural stabilization of the missile body with faster correction

of miss distance. Figure 4 shows the control fins getting

saturated in case of PN law compared to the case of PIGC. It

shows that the objective of minimum distance with minimum

control effort is met. The control values are much below their

saturation limits. To get the miss distance of the same order,

4187

conventional design uses the maximum fin deflection and

finally goes into saturation as shown in Table.I.

TABLE I

SIMULATION RESULTS FOR DIFFERENT INITIAL CONDITIONS

Zero Effort Miss (m) Miss distance(m) Miss distance(m)Cases (with no control) (Three-Loop Design) (PIGC Design)

1 627.71 2.135 1.0182 558.05 2.655 0.7593 522.80 2.060 0.7484 649.17 0.699 0.5095 692.50 2.341 0.552

0 0.5 1 1.5 2 2.5 3 3.5−5

0

5

Time

roll

ra

te

0 0.5 1 1.5 2 2.5 3 3.5−10

0

10

Time

pit

ch

ra

te

0 0.5 1 1.5 2 2.5 3 3.5−10

0

10

Time

ya

w r

ate

PIGC−MPSP

Conventional (three loop)

Fig. 3. Comparison between conventional three loop design and PIGC

0 0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

Time

roll

fin

0 0.5 1 1.5 2 2.5 3 3.5−2

0

2

Time

pit

ch

fin

0 0.5 1 1.5 2 2.5 3 3.5−2

0

2

Time

ya

w f

in

Fig. 4. Comparison between conventional three loop design and PIGC

B. Comparison between One-Loop IGC Design and PIGC

In a One-Loop design, the guidance law and autopilot

design are formulated into a single unified state space

framework. A comparative study has also been carried out

between State dependent Ricatti Equation (SDRE) based one

loop IGC design and PIGC approach[3]. Table II shows

the comparative study of one loop IGC and PIGC in terms

of miss distance and zero effort miss for different cases.

It is evident from Table II that the engagement in PIGC

takes place at a miss distance of the order of less than

0.2m. All the results are normalized trajectories. Figure 5

shows the 3D engagement scenario in inertial frame of

PIGC scheme. Figure 6 represents the magnified view of

1

1.5

2

2.5

1

1.5

2

2.51

1.2

1.4

1.6

1.8

DownrangeCrossrange

He

igh

t

M

T

Fig. 5. Engagement in 3D space for PIGC.

0.570.58

0.590.6

0.61

0.195

0.2

0.205

0.21

0.2151.21

1.22

1.23

1.24

1.25

1.26

Down rangeCross range

He

igh

t

Fig. 6. Magnified view in 3D space for One-loop IGC

3D engagement geometry in case of one loop IGC. It is also

seen in Table II that the objective of minimum distance is not

met. Figure 7 shows the comparative plot of body rates of

conventional IGC and PIGC scheme. Figure 8 represents the

comparative plot of fin deflections obtained through SDRE

and DI controller in PIGC scheme.

TABLE II

SIMULATION RESULTS FOR DIFFERENT INITIAL CONDITIONS

Zero Effort Miss(m) Miss Distance(m) Miss Distance(m)Cases (with no control) (One-Loop IGC) (PIGC)

1 55.27 39.767 0.03482 53.17 32.147 0.17143 27.05 11.88 0.24814 50.49 24.85 0.10975 44.82 14.56 0.2167

C. Simulation Study with Different Initial Condition

Different cases were studied with interceptor and target

initial state perturbations to check the effectiveness of the

PIGC scheme. Body rates were perturbed differently, with

roll rate (p) between ±20o, pitch rate (q) and yaw rate (r)

between ±10o. Random deviation of 2% is considered for

both interceptor and target initial conditions together. Figure

9 shows the family of trajectories of 3D engagement. Figures

10 and 11 represents the body rates and fin deflections of the

interceptor.

4188

0 0.5 1 1.5 2 2.5 3 3.5 4−20

0

20

Time (sec)

Roll

rate

0 0.5 1 1.5 2 2.5 3 3.5 4−5

0

5

Time (sec)

Pitc

h r

ate

0 0.5 1 1.5 2 2.5 3 3.5 4−5

0

5

Time (sec)

Yaw

rate

PIGC

One−Loop IGC

Fig. 7. Body rates

0 0.5 1 1.5 2 2.5 3 3.5−2

0

2

Time (sec)

Ro

ll f

in

0 0.5 1 1.5 2 2.5 3 3.5−1

−0.5

0

Time (sec)

Pitch

fin

0 0.5 1 1.5 2 2.5 3 3.5−2

0

2

Time (sec)

Ya

w f

in

PIGC

One−Loop IGC

Fig. 8. Control deflections

0.51

1.52

2.5

1

1.5

2

2.51

1.2

1.4

1.6

1.8

DownrangeCrossrange

He

igh

t

Fig. 9. Engagement Geometry from various initial conditions

0 0.5 1 1.5 2 2.5 3−5

0

5

Time

roll

ra

te

0 0.5 1 1.5 2 2.5 3−5

0

5

Time

pit

ch

ra

te

0 0.5 1 1.5 2 2.5 3−5

0

5

Time

ya

w r

ate

Fig. 10. Body rates of the interceptor

0 0.5 1 1.5 2 2.5 3−2

0

2

Time

roll

fin

0 0.5 1 1.5 2 2.5 3−5

0

5

Time

pit

ch

fin

0 0.5 1 1.5 2 2.5 3−2

0

2

Time

yaw

fin

Fig. 11. Fin Deflection of the interceptor

V. CONCLUSION

Using powerful nonlinear and optimal control design

methods like MPSP and dynamic inversion, this paper

proposes a new philosophy of partial integrated guidance

and control scheme for high speed targets in the terminal

homing loop of interceptors. The newly proposed PIGC

algorithm yields smaller miss distance with minimum control

effort. PIGC scheme uses Six-DOF nonlinear model for

both guidance and control loop. This intelligent manipu-

lation preserves the inherent time scale separation prop-

erty between the translational and rotational dynamics, and

hence overcomes the deficiency of current IGC designs.

However, it preserves the benefits of the IGC philosophy.

Comparative studies of the proposed PIGC scheme with

three loop conventional design and one loop IGC has also

been performed. Simulations with different initial conditions

shows the advantages of the PIGC scheme. The final miss

distance is well within the lethal range.

REFERENCES

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